8 problems

Alternating-group Beauville structure conjecture. Let $(r,s,t)$ and $(r',s',t')$ be two hyperbolic types. Then almost all alternating groups ${\mathcal{A}}_n$ have an unmixed Beauv…

A characteristically simple group is a finite group isomorphic to $H^k$, where $H$ is finite simple and $k$ is a positive integer. The characteristically simple Beauville group con…

A strongly real Beauville group is a group admitting a Beauville structure satisfying the strong reality condition: there are an automorphism and conjugating elements that simultan…

Let $p$ be a prime. A Beauville $p$-group is a Beauville group whose order is a power of $p$. The Beauville $p^5$-group counting conjecture. For every $p\geq 5$, the number of Beau…

Let $c\tau_1=(r_1,s_1,t_1)$ and $c\tau_2=(r_2,s_2,t_2)$ be two hyperbolic types. Let $G$ be a finite simple classical group of Lie type, with Lie rank sufficiently large. Fixed-typ…

Classical-group Beauville structure conjecture. If the Lie rank of $G$ is large enough, then $G$ admits an unmixed Beauville structure $(x_1,y_1;x_2,y_2)$ such that

Beauville existence conjecture. Every finite nonabelian simple group occurs except $\mathfrak U_{5}$.

Beauville structure conjecture. All finite simple nonabelian groups except $\mathfrak{A}_5$ admit an unmixed Beauville structure.